Rabbit creature's facial features are not moving with the face
When you create your rig and set it as the parent of your model, you have to choose, how the weight for the bones are created. This determines, what parts of your model are affected by which bone.
If your model is one object with several parts, all you need to do is to add all the parts you want to be affected to the vertex group. You can find that under Properties -&gt; Object Data Properties -&gt; Vertex Groups. You can either go to edit mode and set all the verticies you want to be affected as part of the vertec gropu coresponding to your bone, or you can go to Wight paint mode and do int there. In weight paint mode, you can also see the poses. For more complex models you would use weight paint mode, but for your model, i think manualy adding all the verticies in edit mode might work fine.
If your model is split into several objects, you need to set the armature as parent for each object. Then you can adjust the vertex groups and weights the same way.